Polynomial inverse and discrete trace inequalities

Given: Countable Choice, the fixed reference simplex, a shape-regular affine mesh family, polynomial degree $k$, the displayed exponents, and a cell polynomial $v$.

On the fixed finite-dimensional polynomial space, $\mathrm{\mid }v{\mathrm{\mid }}_{W^{\mathrm{\ell },p}}\le C\mathrm{\mid }v{\mathrm{\mid }}_{W^{m,q}}$ when $m\le \mathrm{\ell }$, and every facet $L^p$ norm is bounded by the cell $L^q$ norm, for all $p,q\in [1,\mathrm{\infty }]$. (Polynomial derivative and face norms are controlled on a reference simplex)

Under an affine map, order-$r$ Sobolev seminorms acquire factors $\mathrm{\parallel }A{\mathrm{\parallel }}^r\mathrm{\mid }\det A{\mathrm{\mid }}^{-1\mathrm{/}q}$ on pullback and $\mathrm{\parallel }{A}^{-1}{\mathrm{\parallel }}^r\mathrm{\mid }\det A{\mathrm{\mid }}^{1\mathrm{/}p}$ on pushforward. Its oriented face area vector is $\det (A)A^{-T}\widehat{n}d\widehat{S}$. (Affine pullback of weak derivatives and Sobolev seminorms, Gauss–Green formula on an affine simplex)

Shape regularity gives $\mathrm{\parallel }{A}_K\mathrm{\parallel }\asymp h_K$, $\mathrm{\parallel }{A}_K^{-1}\mathrm{\parallel }\asymp h_K^{-1}$, $\det A_K\asymp h_K^d$, and each facet Jacobian is comparable to $h_K^{d-1}$. (Shape regularity controls affine simplex geometry)

The reference polynomial norm result [F1] uses Sobolev norms whose definition assumes Countable Choice. (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$))